Proposed Electronic Calculator/Chapter 14

by Alan Turing

14. The Design of Delay Lines.

(i) General.- A considerable amount of work has been done on delay lines for R.D.F. purposes. On the whole our problems coincide with the R.D.F. problems but there are a few differences.

(a) Owing to the fact that there will be more than one tank used in the calculator the stability of the delay is of importance. In R.D.F. the delay is allowed to determine the recurrence frequency and the effects of variations in it are thereby eliminated.

(b) In R.D.F. it is required that the delayed signal should not differ from the undelayed by an error signal which is less than 60 dB (say) down on the signal proper. We are less difficult to please in this respect. We only require to be able to distinguish mark from space with a very high probability (e.g. at least 1-10^-32). This requires a high signal to noise ratio, so far as the true random noise and the interference are concerned, but it does not require much as regards hum, frequency distortion and other factors producing unwanted signals of fairly constant amplitude.

Our main concerns then in designing a delay line will be:

(1) To ensure sufficient signal strength that noise does not cause serious effects.

(2) To eliminate or correct frequency and phase distortion sufficiently that we may correctly distinguish mark and space.

(3) To stabilise the delay to within say 0.2 pulse periods.

(4) To eliminate interference.

(5) To provide considerable storage capacity at small cost.

(6) To provide means for setting the crystals sufficiently nearly parallel.

The questions of noise and signal strength are treated in some detail in the following pages. It is found that there is plenty of power available unless either very long lines or very high frequencies are used. The elimination of interference is mainly a matter of shielding and is a very standard radio problem, which in our case is much less serious than usual. Various means have been found by the R.D.F. workers for setting the crystals. Some prefer to machine the whole delay line very accurately, others to provide means for moving the crystals through small angles, e.g. by bending the tank. All are satisfactory.

I list below a number of questions which must be answered in our design of delay lines. In order to fix ideas I have added the most probable answers in brackets after each question.

(1) What liquid should be used in the line? (Either mercury or a water-alcohol mixture.)

(2) Should we use a carrier? If so, of what frequency? (Yes, certainly use a carrier. Frequency should be about 10 Mc/s with water-alcohol mixture, but may be higher if desired when mercury is used).

(3) What should be the clock-pulse frequency? (1 Mc/s).

(4) What should be the dimensions of the crystals? (Diameter might be half that of the tank, e.g. 1 cm. Thickness should be such that the first resonances of the two crystals are two or three megacycles on either side of the carrier, if water-alcohol is used. With mercury the thickness is less critical and may be either as with water-alcohol or may have resonance equal to carrier.

(5) Should the inside of the tank be rough or smooth? (Smooth).

(6) What should be the dimensions of the tank? (Standard tanks to give a delay of 1 ms. should be about 5' long whether water-alcohol or mercury. Diameter ½').

(Keep all the tanks within one degree Fahrenheit in temperature. Correct systematic temperature changes by altering the pulse frequency.)

In order to be able to answer these questions various mathematical problems connected with the delay lines will have to be solved.

(ii) Electromagnetic conversion efficiency. The delay line may best be considered as forming an electrical network of the kind usually (rather misleadingly) described as ‘four-pole’, i.e. a network which has one input current and one input voltage which together determine an output voltage and current. Such a network is described by three complex numbers at each frequency. In the case where there is little coupling between the output and input, which will apply to our problem, we may take these quantities to be the input and output admittances and the ‘transfer admittance’. Strictly speaking we should specify whether the output is open circuit or short circuit when stating the input impedance, but with weak coupling these are effectively the same; similarly for the output impedance. The transfer admittance is the current produced at one end due to unit voltage at the other, and does not depend on which end has the voltage applied to it. In the case of the delay lines the input and output admittances will be effectively the capacities between the crystal electrodes. We need only determine the transfer admittance.

We shall consider the following idealised case. Two crystals of thicknesses d and d' are immersed in a liquid, with their faces perpendicular to the x-axis. The liquid extends to infinity in both the positive and the negative x-directions, and both liquid and crystals extend to infinity in the y and z directions (Fig. 40). The distance between the near side faces of the crystals is ℓ It is assumed that there is considerable attenuation of sound waves over a distance of the order of ℓ but hardly any over a distance of the order of d or d'.

These assumptions are introduced largely with a view to eliminating the possibility of reflections. In practice the reflections would be eliminated by other means. For instance, the infinite liquid on the extreme right and left would be replaced by a short length of liquid in a stub of not very regular shape, so that the reflected waves would not be parallel to the face of the crystal. More likely still, of course, we should have some entirely different medium there.

The physical quantities involved are:

(a) The density ρ. We write ρ for the density of the crystal and ρ₁ for that of the liquid. Likewise a suffix 1 will indicate liquid values throughout.

(b) The pressure p. In the case of the crystal this is understood to mean the xx-component of stress.

(c) The displacement ξ in the x-direction.

(d) The velocity v in the x-direction.

(e) The radian frequency w.

(f) The elasticity η. This is the rate of change of pressure per unit decrease of logarithm of volume due to compression.

(g) The velocity of propagation c.

(h) The mechanical characteristic impedance ζ.

(i) The reciprocal radian wave length β.

(j) The piezo-electric constant ε. This gives the induced pressure due to an electric field strength of unity. This field strength should normally be thought of as in the x-direction, but we shall have to consider the case of a field in the y or z direction briefly also.

These quantities are related by the equations

$c={\sqrt {\eta /\epsilon }},\zeta ={\sqrt {\eta /\rho }},\beta ={\frac {w}{c}},v=iw\xi ,iw\rho v=-{\frac {dp}{dx}},p=-\eta {\frac {d\xi }{dx}}+E\epsilon$

In what follows we assume that all quantities such as p, v, ξ depend on time according to a factor e^iwt, which we omit.

We now consider the ‘transmitting crystal’, which we suppose extends from x = -a to x = a where d = 2a. The solution of the equations will be of form

$p=E\epsilon +B\cos \beta x$

within the crystal, i.e. for |x| < a. Since the pressure is continuous we shall have

$p=(E\epsilon +B\cos \beta a)e^{i\beta _{1}(a-x)}{\mbox{ if }}x>a.$

This gives for the velocity

$v={\frac {1}{w\rho }}\cdot -B\beta \sin \beta x=-B\zeta ^{-1}\sin \beta x{\mbox{ if }}|x|<a$

$v=\zeta _{1}^{-1}(E\epsilon +B\cos \beta a)e^{i\beta _{1}(a-|x|)}{\mbox{ syn }}x{\mbox{ if }}|x|>a$

Continuity of velocity now gives

$B{\Big (}cos\beta a+{\frac {i\zeta _{1}}{\zeta }}sin\beta a{\Big )}=-E\epsilon$

and therefore the velocity at a is

${\frac {-iB\sin \beta a}{\zeta }}={\frac {iE_{2}\sin \beta a}{\zeta \cos \beta a+i\zeta _{1}\sin \beta a}}$

i.e. the velocity at the inside edge of the crystal is

${\frac {iB\epsilon }{\zeta }}\cdot {\frac {1}{\cot {\frac {dw}{2o}}+iu}}$

where $u=\zeta l/\zeta$.

Assuming that the exciting voltage is longitudinal we may say that

${\frac {\mbox{Velocity}}{\mbox{Exciting voltage}}}={\frac {i\epsilon }{\zeta d}}\cdot {\frac {1}{\cot {\frac {dw}{2o}}+iu}}$

The effect of the medium between the two crystals we will not consider just yet. Let us simply assume that

${\frac {\mbox{Velocity at inside edge of receiving crystal}}{\mbox{Velocity at inside edge of transmitting crystal}}}=\vartheta$

We have now to consider the effect of the receiving crystal. Fortunately we can deal with this by the principle of reciprocity. When applied to a mixed electrical and mechanical system this states that the velocity produced at the mechanical end by unit voltage at the electrical end is equal to the current produced at the electrical end by unit force at the mechanical end. Hence

${\frac {\mbox{Current at receiving end}}{\mbox{Force on receiving crystal}}}={\frac {i\epsilon }{d'\zeta }}{\frac {1}{\cot {\frac {d'w}{2o}}+iu}}$.

To these equations we may add that the ratio of force to pressure is the area A’ of the receiving crystal, and that the ratio of pressure to velocity is the mechanical characteristic impedance ζ₁. Combining we obtain

$Y={\mbox{Transfer admittance}}=\vartheta {\frac {A^{'}\eta ^{2}\zeta _{1}}{dd'\zeta ^{2}}}{\frac {1}{{\big (}\cot {\frac {dw}{2o}}+iu{\big )}{\big (}\cot {\frac {d'w}{2o}}+iu{\big )}}}$.

Let us now assume that the input to the valve from the receiving crystal consists of a tuned circuit with a fairly low ‘Q’ as in Fig. 41. Then

Voltage attenuation and phase change factor $={\frac {\mbox{Grid voltage}}{\mbox{Input voltage}}}$

= == Grid voltage Input voltage

where Wg? Lear = 1,

where = Dielectric constant of crystal

A= Attenuation due to viscosity of medium and geometrical causes.

RQw) = = ~ Big f ae 5 ig ore foot — + in on Se) oa ee) 20 20 29 2a 2 ig?

The quantity 7 depends only on the crystal, i.e. on the material of which it is made and its cut and form of excitation. Both £2 nd 1, zt of the dimensions of a pressure. 47 is of the dimensions of an electric field, and may be thought of as a constant electric field which has to be added to the varying field in order that the combination should produce the correct pressure variations, somewhat like the permanent magnet field in a telephone receiver, A typical

value for qin ors HES is 0.004.

Let us now consider the frequency-dependent factor, R(w). The parameter u entering here is the ratio of the characteristic impedances of the crystal and the liquid, It is equal to

$={\frac {\mbox{Velocity of sound in liquid x density of liquid}}{\mbox{Velocity of sound in crystal x density of crystal}}}$ The velocity of sound in the crystal (X-cut quartz) is 5.72 km/sec. and its density is 2.7. The velocity in water is 1.44 km/sec., and the density 1, hence

u(water) = 0.1 abt.

The velocity in mercury is much the same but the density is 13.5. Hence

u(mercury) = 1.3 abt.

These figures suggest that we consider the two cases where u is small and where u is 1. The latter case may be described by saying that the liquid matches the crystal.

It may be assumed for the moment that our object is to make the minimum value of |R(w)| in a certain given band of frequencies as large as possible. If the width of the band is 2Ω and it is centred on w_o and if we ignore the variations in ϑ we shall find that the optimum value of u is of the form ${\mbox{N}}{\frac {\Omega }{w_{0}}}$ where N is some numerical constant probably not too far from 1. The value of Q should be as large as possible. With Ω = 1 Mc/s, w_o = 10 Mc/s this seems to suggest that water (u = 0.1) is very suitable. In practice the differences due to the value of ϑ are more serious than those due to u, and there is in any case plenty of power. We would not in practice take Q as large as we could but would rather try to arrange that |R(w)| was fairly constant throughout the band concerned and arg R(w) fairly linear when plotted against w. If water were used one would probably choose the thicknesses of the crystals and the value of Q to give poles of |R(w)| somewhat as shown in Fig. 41. With this arrangement of the poles the gain corresponding to |R(w)| is 9 dB throughout the range 8 Mc/s and the phase characteristic lies within 5° of the straight line within this range.

With mercury where u is nearly 1 we should put

${\frac {dw_{0}}{2c}}={\frac {\pi }{2}},{\frac {d\ w_{0}}{2c}}={\frac {\pi }{2}},$

and then

${\big |}R(w){\big |}={\frac {2}{\pi }}{\Big (}sin{\frac {\pi }{2}}{\frac {w}{w_{0}}}{\Big )}^{2}{\Bigg |}{\frac {w\ w_{0}}{{\big (}w+w_{0}+{\frac {iw_{0}}{2Q}}{\big )}{\big (}w-w_{0}+{\frac {iw_{0}}{2Q}}{\Big )}}}{\Bigg |}$

We should probably find it desirable to omit the tuned circuit, in which case R(w) would represent a fairly constant loss of 4 dB. One could use a Q of 2 if one wished, giving a gain of 2 dB instead.

We have assumed above that the crystal is longitudinally excited. If it were transversely excited the figures would be much less satisfactory. At the transmitting end a far larger voltage would have to be applied in order to obtain the same field strength, and at the receiving end the stray capacities will have a more serious effect with transverse electrodes, although if the stray capacity were zero transverse electrodes at the receiving end would actually be more efficient.

(iii) Geometrical attenuation.- If a rectangular crystal is crookedly placed in a plane parallel beam, the tilt being such that the one edge of the crystal is advanced in phase by an angle -/~ then the attenuation due to the tilt is -ein 3° ———--. With a square crystal whose side is 1 cm and a frequency of 15 Mc/s this would mean that we get the first zero in the response for a tilt of about 26', The setting is probably not really as critical as this, owing to curvature of the wave fronts. If the crystals are operating in a free medium without the tube this effect is easily estimable and we find that, for crystals sufficiently far apart the allowable angles of tilt are of the order of the angle subtended at one crystal by the other. It has been found experimentally with tubes operating at 15 Mc/s that tilts of the order of half a degree are admissible.

Now let us consider the loss due to boundary effects. We assume

a wave inside the tank of form p = Jol flortpe tint and assume a

boundary condition of form - — = { where we do not know * nor even

whether it is real or complex. The radius of the tank is a, so that

the boundary condition becomes ‘- as Ya. Let the solution Foe) 2 wae) apse? of Ra sy be u(y). Then we have 42 + ( | = -5 and ots) e a oo a, fy 1 z therefore Be Jas Ku Jasco. But since MES is small this ' rare &e y and the 208s in a length £ of the Be Ride we sea pry YY there are many means approximately N/, = ined. tank is abe ay solutions of Se but there is a bounded region of the u plane Fo in which there is always a solution whatever value 5a my have. This means to say that for any boundary condition there is always a mode in

which the attenuation does not exceed T ~s- where UT is some numerical constant.

The value of 7 is about 1.9. It is the largest value of xy such that (x + dy) Jy(x+ iy\/o(x + dy) is pure imaginary and y > 0, 0 < x < 2.4.

Taking Lo/awo = 0.31 (as p. 41) the maximum loss in this mode is 6 dB. We should however probably add a certain amount to allow for the fact that not all of the energy will be in this mode. A total loss of 20 dB would probably not be too small.

(iv) Attenuation in the medium.- The attenuation coefficient is given

by -r-oe where D is the dynamic coefficient of viscosity, i.e., the

ratio of viscosity to density. With water ( = .013 c = 1.44 Km/sec.) at a frequency of 10 megacycles and a delay of 1 ms we have a loss of 12 dB. With mercury under the sane circumstances the loss is only 1 dB. These figures suggest that if water is used the frequency should not be much above 10 Mc/s, but that we can go considerably higher with mercury.

(v) Noise.

Before leaving the subject of attenuation we should verify how much can be tolerated. The limiting factor is the noise, due to thermal agitation and to shot effect in the first amplifying valve. The effect of these is equivalent to an unwanted signal on the grid of the first valve, whose component in a narrow band of width f cycles has an R.M.S. value of

V_N = 4 k T f (R + R_e)

where T is the absolute temperature, k is Boltzmann's constant and R is the resistive component of the impedance of the circuit working into the first valve, including the valve capacities. R_e is a constant for the valve and describes the shot effect for the valve.

In the case that we use mercury and do not tune the input the value of R will be quite negligible in comparison with R_e, which might typically be 1000 ohms. For a pulse frequency of 1 megacycle we must take f = 10⁶ (the theoretical figure is ½10⁶ but this is only attainable with rather peculiar circuits). At normal temperatures 4 k T = 1.6 x 10⁻²⁰ and therefore V_N = 4 μV. In the case that we use water and tune the input, we have R = Q / w(C_x + C_s) at the worst frequency. Assuming w / 2πQ = 2 Mc/s (see Fig. 41) and C_x + C_s = 20 pf and ignoring the fact that the effect will not be so bad at other frequencies, we have V_N = 9 μV.

Now suppose that we wish to make sure that the probability of error is less than p, and that the difference in signal voltage between a digit 0 and a digit 1 is V. Then we shall need

2 dx < p.

(This follows from the fact that a random noise voltage is normally distributed in all its coordinates.) If we put p = 10^-32 we find V/V_N ≥ 24 , V ≥ 0.1 mV.

(vi) Summary of output power results. Summarising the voltage attenuation and noise questions we have:

(a) There is an attenuation factor depending on the material of the crystal and its cut and for quartz typically giving a loss of 48 dB.

(b) There is a factor R depending on the ratio of band width required to carrier frequency, and the matching factor u between crystal and liquid. In practical cases this amounts to gains of 10 dB with water and 2 dB with mercury.

(c) There is a loss factor C_x/C_x + C_s due to stray capacity C_s across the receiving crystal. This might represent a loss of 6 dB.

(d) There is a loss due to the viscosity of the medium. For a water tank with a delay of 1 ms. and a carrier of 10 Mc/s the loss may be 12 dB: with mercury and a carrier of 20 Mc/s it may be 4 dB.

(e) Losses in the walls of the tank. Apparently this should not exceed 10 dB.

(f) The noise voltage may be 4 x 10^-6 volts RMS (mercury) or 9 x 10^-6 volts RMS (water).

(g) The signal voltage (peak to peak) should exceed the noise voltage (RMS) by a factor of 24 for safety.

These figures require input voltages (peak to peak) of 0.2 volts or 4.5 volts with mercury and water respectively. We could quite conveniently put 200 volts on, so that we have 60 dB (or 53 dB) to spare. There is no danger of breaking the crystals when they are operated with so much damping.

(vii) Phase distortion due to reflections from the walls.— We cannot easily treat this problem quantitatively because of lack of information about the boundary conditions and because the ratio of diameter of crystal to diameter of tank is significant. Let us however try to estimate the order of magnitude by assuming the pressure zero on the boundary and considering the gravest mode. In this case the pressure is of form 1 ry ‘ = j eh Pteint where 2a is the diameter of the tank and k1 = 2.4 is the smallest zero of Jo, and 2 ke w 7 Pre sof of tank is <z07 + If we are using carrier working we are chiefly interested in -—t which tums out to be -—Ha=s where wo is the carrier frequency. If we suppose 4a 2.2, then the greatest phase error which is introduced is &y2.92 of 2 oe a® Let us suppose that the greatest admissible error is 0.2 radians, then we must have

Lo Ok fe eed ce

Taking w_o = 10 Mc/s

= 1 Mc/s

c = 1.4 x 10⁵ cm/sec.

= 1.4 x 10² cm.

a = 1 cm.

Then see 2x10 om %o

te spree 032

a ity = 6.95

The situation is thus entirely satisfactory. The carrier frequency could even be halved.

(viii) The choice of medium. In choosing the medium we have to take into account

(a) That a medium with a small characteristic impedance such as water has a slight advantage as regards the factor R(w).

(b) That water is more attenuative than mercury.

(c) That mercury gives wide band widths more easily than water because of closer matching, but that adequate band widths are nevertheless possible with water.

(d) That a water-alcohol mixture can be made to have a zero temperature coefficient of velocity at ordinary temperatures.

On the whole the advantages seem to be slightly on the side of mercury.

(ix) Long lines.— The idea of using delay lines with a long delay, e.g. of the order of 0.1 second, is attractive because of the very large storage capacity that such a line would have. Although the long delay would make these unsuitable for general purposes they would be very suitable for cases where very large amounts of information were to be stored: in the majority of such cases the material is used in a fairly definite order and the long delay does not matter.

However such long lines do not really seem to be very hopeful. In order to reduce the attenuation to reasonable proportions it would be necessary to abandon carrier working, or else to use mercury. In either case we should probably be obliged to make the tank in the form of a bath rather than a tube; in the former case in order to avoid the phase distortion arising from reflections from the walls, and in the latter to economise mercury, using a system of mirrors in the bath. In any case the technique would involve much development work.

We propose therefore to use only tanks with a delay of 1 ms.

(x) Choice of parameters.— Considerations affecting the carrier frequency are:

(a) The higher the carrier frequency the greater the possible band width.

(b) The difficulty of cutting thin crystals, somewhat modified by the absence of necessity of frequency stability.

(c) The attenuation at high frequencies of the sound wave in the liquid.

(d) The difficulty of setting the crystals up sufficiently nearly parallel if the wavelength is short.

(e) The difficulty of amplification at high frequencies.

Of these (a) and (c) are the most important. A reasonable arrangement seems to be to choose a frequency at which the attenuation in the medium is about 15 db.

With the comparatively low frequencies and with wide tanks the setting up difficulty will not be serious. With long lines we should probably not attempt to do temperature correction, but would rephase the output.

Considerations affecting the pulse frequency are:

(a) The limitation of the pulse frequency to a comparatively small fraction of the carrier frequency if water is the transmission medium, and the limitation of this carrier frequency.

(b) The finite reaction times of the valves.

(c) The greater capacity of a line if the frequency is high.

(d) Greater speed of operation of the whole machine if the pulse frequency is high.

(e) Cowardly and irrational doubts as to the feasibility of high frequency working.

If we can ignore (e) the other considerations appear to point to a pulse frequency of about 3 megacycles or even higher. We are however somewhat alarmed by the prospect of even working at 1 megacycle since the difficulty (b) might turn out to be more serious than anticipated.

Considerations affecting the diameter of the tank are:

(a) That the crystals are most conveniently adjusted to be parallel by bending the tanks and that the diameter should therefore not be too large.

(b) That the diameter should be at least large enough to accommodate the crystal.

(c) That small diameters give phase distortion (p. 40).

(d) That with mercury small diameters are economical. At a price of £1 sterling per 1 lb. avoirdupois of mercury a 1 ms. tank of diameter 1" would contain mercury to the value of about £2-2-6.

A diameter of 1" or rather less is usual in R.D.F. tanks and appears reasonable in view of these conditions.

(xi) Temperature control system.— The temperature coefficient of the velocity of propagation in mercury is quite small at 15 Mc/s, being only 0.0003/degree centigrade. This means that if the length of a 1 ms. line is to be correct to within 0.2 ms. then the temperature must be correct to within two-thirds of a degree centigrade.